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The third theorem of similarity theory solves the problem for the
necessary and sufficient conditions determining the similarity of the phenomena
in two systems. According to this theorem similar phenomena are these for
which the uniqueness conditions are similar and the determining criteria
composed of them are equal too.
As already mentioned the similarity theory gives the possibility to
transform the differential equations into functions of similarity criteria by the
following steps:
1. The uniqueness conditions are formulated, i.e. the similarity
constants are given.
2. Each of the terms of the differential equation is multiplied by the
corresponding similarity constant and these constants are taken out before the
differentiation operator, for example:

dx" a" dx"

This transformation leads to obtaining of a system of equations describing a
group of similar phenomena.
3. The coefficients standing before equal terms of the initial and the
transformed equations are equated. The obtained equations or indicators of
similarity correlate the similarity constants.
4. In the obtained equations the similarity constants are replaced by the
respective ratios of values and the criteria of similarity are obtained as
illustrated in the following example.
Let us write Newton's law

(94)

The uniqueness similarity of two similar liquid flows gives the scale
multipliers for the physical values in equation (94). They are as follows: aj- for
the force, a m- for the mass, a w- for the velocity, and a^- for the time. Multiplying
each of the members of the equation by the respective scale multiplier, we
obtain: